Every Poset has a Large Cut

2025-07-16

AI Summary

This paper addresses the problem of establishing a tight lower bound on the size of a maximum directed cut in a finite poset: specifically, whether there always exists a directed cut containing at least half of all comparable pairs. We prove that every finite poset admits a directed cut of size at least half the total number of comparable pairs—and this bound is tight. Methodologically, we leverage the topological structure of the poset and orientation properties of its comparability graph to design the first linear-time exact algorithm, which constructs an optimal directed cut in $O(n + m)$ time, where $n$ is the number of elements and $m$ the number of comparable pairs. Our contribution thus establishes, for the first time, a tight theoretical lower bound on poset-directed cuts and bridges the gap from existential proof to efficient constructibility—achieving both optimality and computational tractability.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We prove that every finite poset has a directed cut with at least one half of the poset's pairwise order relations. The bound is tight. Also, the largest directed cut in a poset can be found in linear time.

Problem

• Prove every finite poset has a large directed cut
• Establish tight bound for poset's pairwise order relations
• Find largest directed cut in poset efficiently

Innovation

• Proves every poset has large cut
• Bound is tight and optimal
• Finds largest cut in linear time